在外加振盪磁場中阻尼磁針的非線性動力學分析

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姓名

許庭瑋(Ting-wei Hsu) 查詢紙本館藏

畢業系所

生物物理研究所

論文名稱

在外加振盪磁場中阻尼磁針的非線性動力學分析
(Numerical and theoretical analysis of the nonlinear dynamics of a damped compass under external oscillatory magnetic field)

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摘要(中)

我們研究了振盪磁場中的阻尼磁針系統，振盪磁場由垂直交錯的兩部分合成，一部分是方向大小固定的磁場B1（可以是地球磁場），另一部分是以正弦形式、振幅為B2的外加振盪磁場。磁針的擺動是複雜的非線性振盪，它會在外磁場振幅增加時，經由週期倍增路徑變成混沌的振盪。系統的運動方程式具有「角度反轉，同時時間平移二分之奇數倍個磁場振盪週期」的對稱性。因為這個對稱性，磁針振盪運動有對稱的週期和混沌吸引子共存。我們用數值方法解微分方程組和畫出相空間圖，檢查當系統的參數改變時，吸引子的性質如何隨參數改變，例如：成對對稱的吸引子是如何出現和合併。有趣的是，我們不只發現了具有相同週期的吸引子成對對稱地出現。在某些參數區間，彼此不對稱的兩個單數週期吸引子也可以共存，以及一對成對對稱的週期二吸引子和一個混沌吸引子的共存也被我們發現。

摘要(英)

We consider a magnetic dipole (compass needle) under a constant magnetic (Earth′s) field and an external sinusoidally oscillating magnetic field (of magnitude B2) that is perpendicular to the former. The angular motion displays complex nonlinear oscillations and undergoes a period-doubling route to chaos. The equation of motion of the system possesses a special symmetry when angle inversion together with time translation of half of the driving period is applied. Due to this symmetry, coexistence of attractors, including symmetric periodic states and symmetric chaotic strange attractors, occurs. The properties of these attractors, such as how the symmetric attractor pairs appear and merge, as revealed by numerical solution of the differential equations and phase portraits, are examined in detail as the parameters of the system change. Interestingly, it is found that in addition to the coexistence of symmetric limit cycle attractor pair (both having the same period state), two different odd-periodic states not related by symmetry, can coexist. In addition, a pair of symmetric period-2 limit cycles and a chaotic attractor can coexist in certain parameter regimes.

關鍵字(中)

★ 非線性動力學
★ 非線性振盪
★ 吸引子共存
★ 對稱性

關鍵字(英)

★ Nonlinear dynamics
★ Nonlinear oscillation
★ Attractors coexisting
★ Symmetry

論文目次

摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
圖目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
表目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
符號說明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
一、緒論 . . . . . . . . . . . . . . . . . . . . . . . . 1
1-1 研究動機 . . . . . . . . . . . . . . . . . . . . . . 2
1-2 理論背景 . . . . . . . . . . . . . . . . . . . . . . 2
1-2-1 非線性動力系統 . . . . . . . . . . . . . . . . . . 2
1-2-2 相空間 . . . . . . . . . . . . . . . . . . . . . . . 3
1-2-3 吸引子 . . . . . . . . . . . . . . . . . . . . . . . 3
1-2-4 Bifurcation（分岔、分歧） . . . . . . . . . . . . 4
1-2-5 Poincaré 映射 . . . . . . . . . . . . . . . . . . . 10
1-2-6 吸引子共存 . . . . . . . . . . . . . . . . . . . . 11
二、文獻探討 . . . . . . . . . . . . . . . . . . . . . . 13
2-1 吸引子共存 . . . . . . . . . . . . . . . . . . . . 13
2-2 磁針、單擺系統 . . . . . . . . . . . . . . . . . . 19
三、研究方法 . . . . . . . . . . . . . . . . . . . . . . 23
3-1 外加振盪磁場中的阻尼磁針系統 . . . . . . . . . 23
3-2 理論分析方法 . . . . . . . . . . . . . . . . . . . 23
3-2-1 分析系統的運動方程式 . . . . . . . . . . . . . . 23
3-2-2 分析系統的對稱性 . . . . . . . . . . . . . . . . 24
3-3 數值分析方法 . . . . . . . . . . . . . . . . . . . 24
3-3-1 使用四階 Runge-Kutta 法及 Euler 法數值模擬
系統 . . . . . . . . . . . . . . . . . . . . . . . . 24
3-3-2 使用 Poincaré 截面和時間平均法幫助判斷磁針
振盪的週期 . . . . . . . . . . . . . . . . . . . . 25
四、研究結果 . . . . . . . . . . . . . . . . . . . . . . 29
4-1 磁針振盪吸引子隨 b2 的變化 . . . . . . . . . . . 29
4-2 系統對稱性和磁針振盪吸引子存在情形的分析 . 42
4-2-1 只有成對對稱吸引子的共存 . . . . . . . . . . . 42
4-2-2 只有自我對稱吸引子的共存 . . . . . . . . . . . 42
4-2-3 成對對稱吸引子和自我對稱吸引子的共存 . . . . 43
4-2-4 系統不允許偶數週期的自我對稱吸引子 . . . . . 43
4-3 Basins of attraction 的性質 . . . . . . . . . . . . 44
4-3-1 自我相似的結構 . . . . . . . . . . . . . . . . . . 44
4-3-2 與系統對稱性的關係 . . . . . . . . . . . . . . . 45
4-3-3 隨參數 b2 的變化 . . . . . . . . . . . . . . . . . 47
五、討論、結論與未來展望 . . . . . . . . . . . . . . 49
5-1 討論 . . . . . . . . . . . . . . . . . . . . . . . . 49
5-1-1 與磁針、單擺系統文獻的比較 . . . . . . . . . . 49
5-1-2 與其他吸引子共存系統文獻的比較 . . . . . . . . 50
5-2 結論 . . . . . . . . . . . . . . . . . . . . . . . . 50
5-3 未來展望 . . . . . . . . . . . . . . . . . . . . . . 51
參考文獻 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
附錄一 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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指導教授

黎璧賢、陳志強(Pik-yin Lai Chi-keung Chan)

審核日期

2016-1-28

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